Kinematics of Rigid Bodies

While particle kinematics ignores rotational effects, rigid body kinematics accounts for both the translation and rotation of bodies. A rigid body is idealized as a system of particles wherein the distance between any two specific particles remains perfectly constant under applied loads.

Types of Rigid Body Motion

Rigid Body Motion Categories

  1. Translation: Any straight line segment drawn on the body remains parallel to its original orientation throughout the motion. All particles in the body have the exact same velocity and acceleration.
    • Rectilinear Translation: Paths of particles are straight lines.
    • Curvilinear Translation: Paths of particles are congruent curves.
  2. Rotation About a Fixed Axis: All particles move in circular paths centered on the axis of rotation, except for those lying on the axis (which have zero velocity).
  3. General Plane Motion: A combination of translation and rotation. The body undergoes a displacement consisting of a translation within a reference plane and a rotation about an axis perpendicular to that plane.
Interact with the simulation below to explore rigid body translation.

Rigid Body Translation Simulation

Notice how the line connecting points A and B remains parallel to its original orientation throughout the motion.

Rotation About a Fixed Axis

When a rigid body rotates about a fixed axis, its motion is described by angular parameters.

Angular Kinematic Variables

  • Angular Position (θ\theta): The angle defining the position of a line on the body, usually measured in radians.
  • Angular Velocity (ω\omega): The time rate of change of angular position. ω=dθdt\omega = \frac{d\theta}{dt}
  • Angular Acceleration (α\alpha): The time rate of change of angular velocity. α=dωdt=d2θdt2\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}
Similar to linear motion, eliminating dtdt yields:
α dθ=ω dω\alpha \, d\theta = \omega \, d\omega
If angular acceleration αc\alpha_c is constant, we obtain angular equations entirely analogous to linear constant acceleration formulas:

Constant Angular Acceleration Equations

ω=ω0+αct\omega = \omega_0 + \alpha_c tθ=θ0+ω0t+12αct2\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha_c t^2ω2=ω02+2αc(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha_c(\theta - \theta_0)

Velocity and Acceleration of a Point

Point Kinematics in Rotation

For a point PP located at a distance rr from the axis of rotation:
  • Velocity (vv): v=ωrv = \omega r (direction is tangent to the circular path).
  • Tangential Acceleration (ata_t): at=αra_t = \alpha r (represents the change in the magnitude of velocity).
  • Normal Acceleration (ana_n): an=ω2r=v2/ra_n = \omega^2 r = v^2/r (directed towards the axis of rotation).

Relative Motion Analysis Using Translating Axes

A powerful method for analyzing general plane motion is to break the motion down into two parts: a translation of a chosen base point, and a pure rotation about that base point. This relies on non-rotating (translating) axes attached to the base point.

Relative Velocity and Acceleration

Let points A and B lie on the same rigid body undergoing general plane motion. A translating reference frame is attached to base point A.
Relative Velocity: The velocity of B equals the velocity of A plus the velocity of B relative to A due to rotation.
vB=vA+ω×rB/A\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{B/A}
Where ω×rB/A\mathbf{\omega} \times \mathbf{r}_{B/A} gives the relative tangential velocity (vB/A)(\mathbf{v}_{B/A}). Its magnitude is vB/A=ωrB/Av_{B/A} = \omega r_{B/A}, and its direction is perpendicular to the vector rB/A\mathbf{r}_{B/A}.
Relative Acceleration: The acceleration of B equals the acceleration of A plus the relative acceleration of B with respect to A.
aB=aA+α×rB/A−ω2rB/A\mathbf{a}_B = \mathbf{a}_A + \mathbf{\alpha} \times \mathbf{r}_{B/A} - \omega^2 \mathbf{r}_{B/A}
The relative acceleration consists of two components:
  • Tangential: at=α×rB/A\mathbf{a}_t = \mathbf{\alpha} \times \mathbf{r}_{B/A} (magnitude αrB/A\alpha r_{B/A}, perpendicular to rB/A\mathbf{r}_{B/A}).
  • Normal: an=−ω2rB/A\mathbf{a}_n = - \omega^2 \mathbf{r}_{B/A} (magnitude ω2rB/A\omega^2 r_{B/A}, directed from B towards A).

Relative Motion Analysis Using Rotating Axes

While translating axes are sufficient for simple problems where points A and B lie on the same rigid body, many mechanisms involve sliding contacts on rotating links. In these cases, it's necessary to express relative motion using a coordinate system (x,y)(x,y) that is attached to a rotating body.

Coriolis Acceleration in Planar Motion

When a particle moves with a velocity vrel\mathbf{v}_{rel} relative to a frame rotating with angular velocity ω\mathbf{\omega}, an extra acceleration component arises, known as the Coriolis acceleration (ac\mathbf{a}_c).
vB=vA+Ω×rB/A+(vB/A)xyz\mathbf{v}_B = \mathbf{v}_A + \mathbf{\Omega} \times \mathbf{r}_{B/A} + (\mathbf{v}_{B/A})_{xyz}aB=aA+Ω˙×rB/A+Ω×(Ω×rB/A)+2Ω×(vB/A)xyz+(aB/A)xyz\mathbf{a}_B = \mathbf{a}_A + \dot{\mathbf{\Omega}} \times \mathbf{r}_{B/A} + \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}_{B/A}) + 2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz} + (\mathbf{a}_{B/A})_{xyz}
Where:
  • Ω\mathbf{\Omega} and Ω˙\dot{\mathbf{\Omega}} are the angular velocity and acceleration of the rotating frame.
  • (vB/A)xyz(\mathbf{v}_{B/A})_{xyz} is the velocity of point BB relative to the moving frame.
  • (aB/A)xyz(\mathbf{a}_{B/A})_{xyz} is the acceleration of point BB relative to the moving frame.
  • 2Ω×(vB/A)xyz2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz} is the Coriolis Acceleration.

General Plane Motion: Instantaneous Center of Zero Velocity (IC)

General plane motion can be viewed at any specific instant as a pure rotation about a single, unique axis. The point where this axis intersects the plane of motion is called the Instantaneous Center of Zero Velocity (IC).

Important

The IC is a point in the plane of motion that momentarily has zero velocity (vIC=0v_{IC} = 0). By locating the IC, the velocity of any other point AA on the body can be found simply as vA=ω⋅rA/ICv_A = \omega \cdot r_{A/IC}, with the direction of vAv_A perpendicular to the line connecting AA and the IC.
Note: The IC only has zero velocity at that instant. Its acceleration is generally not zero. Therefore, the IC method is strictly for velocity analysis, not acceleration analysis.
Interact with the simulation below to explore the Instantaneous Center of Zero Velocity.

Instantaneous Center of Rotation (IC)

Angular Velocity (ω\omega)2 rad/s
Radius (rr)2 m

The contact point with the ground (Red Dot) is the Instantaneous Center (IC) because its velocity is zero.

Velocity of any point is perpendicular to the line connecting it to the IC, with magnitude v=d⋅ωv = d \cdot \omega.

Key Takeaways
  • Rotation About a Fixed Axis: Governed by equations ω=dθ/dt\omega = d\theta/dt and α=dω/dt\alpha = d\omega/dt.
  • Point Acceleration: Has normal (an=ω2ra_n = \omega^2 r) and tangential (at=αra_t = \alpha r) components.
  • General Plane Motion is the superposition of translation of a reference point and rotation about that point.
  • Relative Velocity and Acceleration use vector equations (vB=vA+ω×r\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}) to relate points on the same rigid body.
  • Instantaneous Center (IC): A powerful technique to find velocities by treating general plane motion instantaneously as pure rotation about the IC.
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